Mercurial > hg > Members > kono > Proof > ZF-in-agda

comparison src/filter.agda @ 1138:dd18bb8d2893

Find changesets by keywords (author, files, the commit message), revision number or hash, or revset expression.
...
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Fri, 13 Jan 2023 13:03:45 +0900
parents d618788852e5
children 4517d0721b59
comparison
equal deleted inserted replaced
1137:d618788852e5 1138:dd18bb8d2893
300 300
301 open zorn O _⊂_ PO 301 open zorn O _⊂_ PO
302 302
303 open import Relation.Binary.Structures 303 open import Relation.Binary.Structures
304 304
305 record is-filter { L P : HOD } (LP : L ⊆ Power P) (filter : Ordinal ) : Set n where 305 record IsFilter { L P : HOD } (LP : L ⊆ Power P) (filter : Ordinal ) : Set n where
306 field 306 field
307 f⊆L : (* filter) ⊆ L 307 f⊆L : (* filter) ⊆ L
308 filter1 : { p q : Ordinal } → odef L q → odef (* filter) p → (* p) ⊆ (* q) → odef (* filter) q 308 filter1 : { p q : Ordinal } → odef L q → odef (* filter) p → (* p) ⊆ (* q) → odef (* filter) q
309 filter2 : { p q : Ordinal } → odef (* filter) p → odef (* filter) q → odef L (& ((* p) ∩ (* q))) → odef (* filter) (& ((* p) ∩ (* q))) 309 filter2 : { p q : Ordinal } → odef (* filter) p → odef (* filter) q → odef L (& ((* p) ∩ (* q))) → odef (* filter) (& ((* p) ∩ (* q)))
310 proper : ¬ (odef (* filter ) o∅) 310 proper : ¬ (odef (* filter ) o∅)
311 311
312 -- all filter contains F 312 -- all filter contains F
313 F⊆X : { L P : HOD } (LP : L ⊆ Power P) → Filter {L} {P} LP → HOD 313 F⊆X : { L P : HOD } (LP : L ⊆ Power P) → Filter {L} {P} LP → HOD
314 F⊆X {L} {P} LP F = record { od = record { def = λ x → is-filter {L} {P} LP x ∧ ( filter F ⊆ * x) } ; odmax = osuc (& L) 314 F⊆X {L} {P} LP F = record { od = record { def = λ x → IsFilter {L} {P} LP x ∧ ( filter F ⊆ * x) } ; odmax = osuc (& L)
315 ; <odmax = λ {x} lt → fx00 lt } where 315 ; <odmax = λ {x} lt → fx00 lt } where
316 fx00 : {x : Ordinal } → is-filter LP x ∧ filter F ⊆ * x → x o< osuc (& L) 316 fx00 : {x : Ordinal } → IsFilter LP x ∧ filter F ⊆ * x → x o< osuc (& L)
317 fx00 {x} lt = subst (λ k → k o< osuc (& L)) &iso ( ⊆→o≤ (is-filter.f⊆L (proj1 lt )) ) 317 fx00 {x} lt = subst (λ k → k o< osuc (& L)) &iso ( ⊆→o≤ (IsFilter.f⊆L (proj1 lt )) )
318 318
319 F→Maximum : {L P : HOD} (LP : L ⊆ Power P) → ({p : HOD} → L ∋ p → L ∋ ( P ＼ p)) → ({p q : HOD} → L ∋ p → L ∋ q → L ∋ (p ∩ q)) 319 F→Maximum : {L P : HOD} (LP : L ⊆ Power P) → ({p : HOD} → L ∋ p → L ∋ ( P ＼ p)) → ({p q : HOD} → L ∋ p → L ∋ q → L ∋ (p ∩ q))
320 → (F : Filter {L} {P} LP) → o∅ o< & L → {y : Ordinal } → odef (filter F) y → (¬ (filter F ∋ od∅)) → MaximumFilter {L} {P} LP F 320 → (F : Filter {L} {P} LP) → o∅ o< & L → {y : Ordinal } → odef (filter F) y → (¬ (filter F ∋ od∅)) → MaximumFilter {L} {P} LP F
321 F→Maximum {L} {P} LP NEG CAP F 0<L 0<F Fprop = record { mf = mf ; F⊆mf = ? 321 F→Maximum {L} {P} LP NEG CAP F 0<L {y} 0<F Fprop = record { mf = mf ; F⊆mf = ?
322 ; proper = subst (λ k → ¬ ( odef (filter mf ) k)) (sym ord-od∅) ( is-filter.proper imf) ; is-maximum = {!!} } where 322 ; proper = subst (λ k → ¬ ( odef (filter mf ) k)) (sym ord-od∅) ( IsFilter.proper imf) ; is-maximum = {!!} } where
323 FX : HOD 323 FX : HOD
324 FX = F⊆X {L} {P} LP F 324 FX = F⊆X {L} {P} LP F
325 FX∋F : odef FX (& (filter F))
326 oF = & (filter F) 325 oF = & (filter F)
327 mf11 : { p q : Ordinal } → odef L q → odef (* oF) p → (* p) ⊆ (* q) → odef (* oF) q 326 mf11 : { p q : Ordinal } → odef L q → odef (* oF) p → (* p) ⊆ (* q) → odef (* oF) q
328 mf11 {p} {q} Lq Fp p⊆q = subst₂ (λ j k → odef j k ) (sym *iso) &iso ( filter1 F (subst (λ k → odef L k) (sym &iso) Lq) 327 mf11 {p} {q} Lq Fp p⊆q = subst₂ (λ j k → odef j k ) (sym *iso) &iso ( filter1 F (subst (λ k → odef L k) (sym &iso) Lq)
329 (subst₂ (λ j k → odef j k ) *iso (sym &iso) Fp) p⊆q ) 328 (subst₂ (λ j k → odef j k ) *iso (sym &iso) Fp) p⊆q )
330 mf12 : { p q : Ordinal } → odef (* oF) p → odef (* oF) q → odef L (& ((* p) ∩ (* q))) → odef (* oF) (& ((* p) ∩ (* q))) 329 mf12 : { p q : Ordinal } → odef (* oF) p → odef (* oF) q → odef L (& ((* p) ∩ (* q))) → odef (* oF) (& ((* p) ∩ (* q)))
331 mf12 {p} {q} Fp Fq Lpq = subst (λ k → odef k (& ((* p) ∩ (* q))) ) (sym *iso) 330 mf12 {p} {q} Fp Fq Lpq = subst (λ k → odef k (& ((* p) ∩ (* q))) ) (sym *iso)
332 ( filter2 F (subst₂ (λ j k → odef j k ) *iso (sym &iso) Fp) (subst₂ (λ j k → odef j k ) *iso (sym &iso) Fq) Lpq) 331 ( filter2 F (subst₂ (λ j k → odef j k ) *iso (sym &iso) Fp) (subst₂ (λ j k → odef j k ) *iso (sym &iso) Fq) Lpq)
332 FX∋F : odef FX (& (filter F))
333 FX∋F = ⟪ record { f⊆L = subst (λ k → k ⊆ L) (sym *iso) (f⊆L F) ; filter1 = mf11 ; filter2 = mf12 333 FX∋F = ⟪ record { f⊆L = subst (λ k → k ⊆ L) (sym *iso) (f⊆L F) ; filter1 = mf11 ; filter2 = mf12
334 ; proper = subst₂ (λ j k → ¬ (odef j k) ) (sym *iso) ord-od∅ Fprop } 334 ; proper = subst₂ (λ j k → ¬ (odef j k) ) (sym *iso) ord-od∅ Fprop }
335 , subst (λ k → filter F ⊆ k ) (sym *iso) ( λ x → x ) ⟫ 335 , subst (λ k → filter F ⊆ k ) (sym *iso) ( λ x → x ) ⟫
336 SUP⊆ : (B : HOD) → B ⊆ FX → IsTotalOrderSet B → SUP FX B 336 SUP⊆ : (B : HOD) → B ⊆ FX → IsTotalOrderSet B → SUP FX B
337 SUP⊆ B B⊆FX OB = record { sup = Union B ; isSUP = record { ax = mf13 ; x≤sup = ? } } where 337 SUP⊆ B B⊆FX OB with trio< (& B) o∅
338 ... | tri< a ¬b ¬c = ⊥-elim (¬x<0 a )
339 ... | tri≈ ¬a b ¬c = record { sup = filter F ; isSUP = record { ax = FX∋F ; x≤sup = λ {y} zy → ⊥-elim (o<¬≡ (sym b) (∈∅< zy)) } }
340 ... | tri> ¬a ¬b 0<B = record { sup = Union B ; isSUP = record { ax = mf13 ; x≤sup = ? } } where
341 mf20 : HOD
342 mf20 = ODC.minimal O B (λ eq → (o<¬≡ (cong (&) (sym (==→o≡ eq))) (subst (λ k → k o< & B) (sym ord-od∅) 0<B )))
343 mf18 : odef B (& mf20 )
344 mf18 = ODC.x∋minimal O B (λ eq → (o<¬≡ (cong (&) (sym (==→o≡ eq))) (subst (λ k → k o< & B) (sym ord-od∅) 0<B )))
338 mf16 : Union B ⊆ L 345 mf16 : Union B ⊆ L
339 mf16 record { owner = b ; ao = Bb ; ox = bx } = is-filter.f⊆L ( proj1 ( B⊆FX Bb )) bx 346 mf16 record { owner = b ; ao = Bb ; ox = bx } = IsFilter.f⊆L ( proj1 ( B⊆FX Bb )) bx
340 mf17 : {p q : Ordinal} → odef L q → odef (* (& (Union B))) p → * p ⊆ * q → odef (* (& (Union B))) q 347 mf17 : {p q : Ordinal} → odef L q → odef (* (& (Union B))) p → * p ⊆ * q → odef (* (& (Union B))) q
341 mf17 {p} {q} Lq bp p⊆q with subst (λ k → odef k p ) *iso bp 348 mf17 {p} {q} Lq bp p⊆q with subst (λ k → odef k p ) *iso bp
342 ... | record { owner = owner ; ao = ao ; ox = ox } = subst (λ k → odef k q) (sym *iso) 349 ... | record { owner = owner ; ao = ao ; ox = ox } = subst (λ k → odef k q) (sym *iso)
343 record { owner = ? ; ao = ? ; ox = ? } 350 record { owner = owner ; ao = ao ; ox = IsFilter.filter1 mf30 Lq ox p⊆q } where
344 mf14 : is-filter LP (& (Union B)) 351 mf30 : IsFilter {L} {P} LP owner
345 mf14 = record { f⊆L = subst (λ k → k ⊆ L) (sym *iso) mf16 ; filter1 = mf17 ; filter2 = ? ; proper = ? } 352 mf30 = proj1 ( B⊆FX ao )
353 mf31 : {p q : Ordinal} → odef (* (& (Union B))) p → odef (* (& (Union B))) q → odef L (& ((* p) ∩ (* q))) → odef (* (& (Union B))) (& ((* p) ∩ (* q)))
354 mf31 {p} {q} bp bq Lpq with subst (λ k → odef k p ) *iso bp | subst (λ k → odef k q ) *iso bq
355 ... | record { owner = bp ; ao = Bbp ; ox = bbp } | record { owner = bq ; ao = Bbq ; ox = bbq }
356 with OB (subst (λ k → odef B k) (sym &iso) Bbp) (subst (λ k → odef B k) (sym &iso) Bbq)
357 ... | tri< bp⊂bq ¬b ¬c = ?
358 ... | tri≈ ¬a bq=bp ¬c = ?
359 ... | tri> ¬a ¬b bq⊂bp = ?
360 mf14 : IsFilter LP (& (Union B))
361 mf14 = record { f⊆L = subst (λ k → k ⊆ L) (sym *iso) mf16 ; filter1 = mf17 ; filter2 = mf31 ; proper = ? }
346 mf15 : filter F ⊆ Union B 362 mf15 : filter F ⊆ Union B
347 mf15 {x} Fx = record { owner = ? ; ao = ? ; ox = subst (λ k → odef k x) (sym *iso) Fx } 363 mf15 {x} Fx = record { owner = & mf20 ; ao = mf18 ; ox = subst (λ k → odef k x) (sym *iso) (mf22 Fx) } where
364 mf22 : odef (filter F) x → odef mf20 x
365 mf22 Fx = subst (λ k → odef k x) *iso ( proj2 (B⊆FX mf18) Fx )
348 mf13 : odef FX (& (Union B)) 366 mf13 : odef FX (& (Union B))
349 mf13 = ⟪ mf14 , subst (λ k → filter F ⊆ k ) (sym *iso) mf15 ⟫ 367 mf13 = ⟪ mf14 , subst (λ k → filter F ⊆ k ) (sym *iso) mf15 ⟫
350 mx : Maximal FX 368 mx : Maximal FX
351 mx = Zorn-lemma (∈∅< FX∋F) SUP⊆ 369 mx = Zorn-lemma (∈∅< FX∋F) SUP⊆
352 imf : is-filter {L} {P} LP (& (Maximal.maximal mx)) 370 imf : IsFilter {L} {P} LP (& (Maximal.maximal mx))
353 imf = proj1 (Maximal.as mx) 371 imf = proj1 (Maximal.as mx)
354 mf00 : (* (& (Maximal.maximal mx))) ⊆ L 372 mf00 : (* (& (Maximal.maximal mx))) ⊆ L
355 mf00 = is-filter.f⊆L imf 373 mf00 = IsFilter.f⊆L imf
356 mf01 : { p q : HOD } → L ∋ q → (* (& (Maximal.maximal mx))) ∋ p → p ⊆ q → (* (& (Maximal.maximal mx))) ∋ q 374 mf01 : { p q : HOD } → L ∋ q → (* (& (Maximal.maximal mx))) ∋ p → p ⊆ q → (* (& (Maximal.maximal mx))) ∋ q
357 mf01 {p} {q} Lq Fq p⊆q = is-filter.filter1 imf Lq Fq 375 mf01 {p} {q} Lq Fq p⊆q = IsFilter.filter1 imf Lq Fq
358 (λ {x} lt → subst (λ k → odef k x) (sym *iso) ( p⊆q (subst (λ k → odef k x) *iso lt ) )) 376 (λ {x} lt → subst (λ k → odef k x) (sym *iso) ( p⊆q (subst (λ k → odef k x) *iso lt ) ))
359 mf02 : { p q : HOD } → (* (& (Maximal.maximal mx))) ∋ p → (* (& (Maximal.maximal mx))) ∋ q → L ∋ (p ∩ q) 377 mf02 : { p q : HOD } → (* (& (Maximal.maximal mx))) ∋ p → (* (& (Maximal.maximal mx))) ∋ q → L ∋ (p ∩ q)
360 → (* (& (Maximal.maximal mx))) ∋ (p ∩ q) 378 → (* (& (Maximal.maximal mx))) ∋ (p ∩ q)
361 mf02 {p} {q} Fp Fq Lpq = subst₂ (λ j k → odef (* (& (Maximal.maximal mx))) (& (j ∩ k ))) *iso *iso ( 379 mf02 {p} {q} Fp Fq Lpq = subst₂ (λ j k → odef (* (& (Maximal.maximal mx))) (& (j ∩ k ))) *iso *iso (
362 is-filter.filter2 imf Fp Fq (subst₂ (λ j k → odef L (& (j ∩ k ))) (sym *iso) (sym *iso) Lpq )) 380 IsFilter.filter2 imf Fp Fq (subst₂ (λ j k → odef L (& (j ∩ k ))) (sym *iso) (sym *iso) Lpq ))
363 mf : Filter {L} {P} LP 381 mf : Filter {L} {P} LP
364 mf = record { filter = * (& (Maximal.maximal mx)) ; f⊆L = mf00 382 mf = record { filter = * (& (Maximal.maximal mx)) ; f⊆L = mf00
365 ; filter1 = mf01 383 ; filter1 = mf01
366 ; filter2 = mf02 } 384 ; filter2 = mf02 }
367 385